The University of Southampton
Courses

# ISVR3073 Theoretical and Computational Acoustics

## Module Overview

This module introduces students to mathematical and numerical methods to solve practical problems in acoustics. It provides a self-contained review and derivation of the equations of linear acoustics in the time and frequency domains. Mathematical modelling of sound fields generated by complex source distributions is introduced. This leads to more advanced mathematical methods commonly used in acoustics such as the acoustic Green function and integral solutions of the acoustic wave equation. The numerical methods which are covered in the course are available as commercial software packages but the underpinning theory and analysis is discussed in sufficient technical detail to serve as a starting point for those seeking to apply or extend them to research problems.

### Aims and Objectives

#### Learning Outcomes

##### Knowledge and Understanding

Having successfully completed this module, you will be able to demonstrate knowledge and understanding of:

• The equations that govern the propagation of sound in a stationary medium. [SMm5]
• Boundary conditions for practical acoustic problems. [SM5m]
• Advanced mathematical methods associated with modelling sound fields generated by complex source distributions. [SM5m]
• Integral solutions of the inhomogeneous Helmholtz equation using the acoustic Green function. [SM2p]
• Some background theories, features and limitations of Finite Element and Boundary Element Methods in the frequency domain for acoustics. [SM5m]
• Trade-off between computational cost and accuracy. [SM5m]
##### Subject Specific Intellectual and Research Skills

Having successfully completed this module you will be able to:

• Formulate solutions to predict sound fields generated by complex source distributions. [SM2m]
• Recognise more advanced mathematical methods in analytical acoustics. [SM1p]
• Assess the suitability of different numerical methods for some practical acoustical problems. [SM5m]
• Validate a numerical acoustics code against a relevant benchmark acoustic problem. [EA3p]
##### Transferable and Generic Skills

Having successfully completed this module you will be able to:

• Apply critical analysis and evaluation skills. [SM3p]
• Read, understand, and interpret scientific texts. [EP4i]
• Synthesise information from a range of sources. [EP4i]
• Communicate clearly in written reports. [D6p]
##### Subject Specific Practical Skills

Having successfully completed this module you will be able to:

• Recognise that Green function theory can be used for solving partial differential equations. [SM2i]
• Understand user documentation for commercial acoustic codes and use relevant tools to create simple computational models, perform analysis and post-process results. [SM5m]
• Determine the mesh requirements and boundary conditions for simulating target acoustic problems. [EA3p]
• Apply the numerical methods presented in the course to problems in acoustics. [EA3p]

### Syllabus

Indicative content: - Revision of fluid dynamics - Derivation of equations for linear acoustics. Wave equation - Time-harmonic acoustics. Complex notation and the Helmholtz equation - Finite Element Method for the Helmholtz problem: 1-D elements - Numerical dispersion and dissipation, the pollution effect. - Finite Element Method for the Helmholtz problem: 2D and 3D elements. - Acoustic sources. Inhomogeneous wave and Helmholtz equations - The acoustic Green function - Integral solutions of the inhomogeneous Helmholtz equation - Boundary Element Method for the Helmholtz problems in 2D and 3D fields. - A range of benchmark examples/applications in physical acoustics

### Learning and Teaching

#### Teaching and learning methods

The course will be delivered by using a mixture of interactive lecture/tutorial sessions. These sessions will be used to present the theory and worked examples. Lecture notes will be available in electronic format on Blackboard. Problems sheets will be provided which contain exercises similar to the worked examples presented during the lectures. Solutions to the exercises will be provided on Blackboard. Solutions to problems will be covered at tutorial sessions. Revision lectures will be given at the end of the course to prepare students for the exam. A summative coursework assignment will require the students to solve acoustics problems by writing simple programmes or by using commercial acoustics software.

TypeHours
Follow-up work24
Revision24
Preparation for scheduled sessions24
Tutorial6
Lecture30
Total study time150

Recommended textbooks. No single text book is available which will cover all the material from the module. Useful texts are: Foundations of Engineering Acoustics by Frank Fahy. Publisher: Academic Press Ltd. Library class mark: TA365 FAH Active Control of Sound by P.A. Nelson and S.J. Elliott. Publisher: Academic Press Ltd. Library class mark: QC247 NEL Sound and Sources of Sound by A.P. Dowling and J.E. Ffowcs Williams. Publisher: Ellis Horwood Ltd. Library class mark: QC225 DOW Lecture notes on the Mathematics of Acoustics Edited by M.C.M. Wright. Publisher: Imperial College Press. Library class mark: QC 223 WRI Acoustics: an introduction to its physical principles and applications by A.D. Pierce. Publisher: McGraw-Hill, Inc. Library class mark: QC225 PIE Acoustics and Aerodynamic Sound by M. Howe. Publisher: Cambridge University Press. Library class mark: TA 365 HOW The following unpublished text is available to download without charge for academic use only: An Introduction to Acoustics by S.W. Rienstra and A. Hirschberg, Eindhoven University of Technology https://www.win.tue.nl/~sjoerdr/papers/boek.pdf The finite Element Method, volume 1. The basis by O C Zienkiewicz and R L Taylor. Butterworth-Heinemann, (Oxford) 2000 Finite element analysis of acoustic scattering. by Frank Ihlenburg. Springer, NY, 1998. Boundary element acoustics, Fundamentals and Computer Codes by T. W. Wu (ed). WIT press, 2000. Computational Simulation in Architectural and Environmental Acoustics by T.Sakagami, S. Sakamoto and T. Otsuro (eds). Springer, Tokyo, 2014. Fast Multipole Boundary Element Method. Theory and Applications in Engineering. by Yijun Liu. CUP 2009.

### Assessment

#### Summative

MethodPercentage contribution
Coursework  () 30%
Exam  (120 minutes) 70%

#### Repeat Information

Repeat type: Internal & External